Solve x/(x^4)^6/3

$$\frac { x } { ( x ^ { 4 } ) ^ { 6 / 3 } }$$

Evaluate

$$\frac{1}{x^{7}}$$

Solution Steps

Divide 6 by 3 to get 2.

$$\frac{x}{\left(x^{4}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{x}{x^{8}}$$

Rewrite x^{8} as xx^{7}. Cancel out x in both numerator and denominator.

$$\frac{1}{x^{7}}$$

Differentiate w.r.t. x

$$-\frac{7}{x^{8}}$$

Steps Using Chain Rule

Divide 6 by 3 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x}{\left(x^{4}\right)^{2}})$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x}{x^{8}})$$

Rewrite x^{8} as xx^{7}. Cancel out x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{7}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(x^{7}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{7})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(x^{7}\right)^{-2}\times 7x^{7-1}$$

Simplify.

$$-7x^{6}\left(x^{7}\right)^{-2}$$